Theorem euabsn2

Description: Another way to express existential uniqueness of a wff: its class abstraction is a singleton. (Contributed by Mario Carneiro, 14-Nov-2016)

	Ref	Expression
Assertion	euabsn2	|- ( E! x ph <-> E. y { x | ph } = { y } )

Proof

Step	Hyp	Ref	Expression
1		eu6	|- ( E! x ph <-> E. y A. x ( ph <-> x = y ) )
2		absn	|- ( { x | ph } = { y } <-> A. x ( ph <-> x = y ) )
3	2	exbii	|- ( E. y { x | ph } = { y } <-> E. y A. x ( ph <-> x = y ) )
4	1 3	bitr4i	|- ( E! x ph <-> E. y { x | ph } = { y } )